Here, we will also show how the model can be extended with edge features and global communication between the agents. In practice, we always include global agent communication. If the given dataset does not make use of edge features, we use a simplified model where the pre-aggregation update functions \(\phi _{N(v) \rightarrow v}\)  and \(\phi _{a \rightarrow v}\)  are set to identity. As MLPs are universal function approximators, previous MLP can already directly approximate \(\phi _{N(v) \rightarrow v}\)  and \(\phi _{a \rightarrow v}\) , thus we do not lose expressiveness by excluding them [1]}. One could also imagine, that providing the agents with the total number of nodes in the graph could help. If the task, for example, is to estimate the density of triangles in the graph, by observing just a fraction of it. Such conditioning can be trivially achieved by including \(n\)  as input to every update function. However, in our experiments, we did not consider such augmentation.
